1 Introduction

A pictorial illustration of conventional cryptography Lars Ramkilde Knudsen March 1993 Abstract In this pap er we consider conventional cryptosystems. We illustrate the di erences b etween substitution, transp osition and pro duct ciphers by showing encryptions of a highly redundant cleartext, a p or- trait. Also it is pictorially demonstrated that no conventional cryp- tosystem in its basic mo de provides sucient security for redundant cleartexts. 1 Intro duction The history of cryptography is long and go es back at least 4,000 years to the Egyptians, who used hieroglyphic co des for inscription on tombs [2]. Since then many ciphers have b een develop ed and used. Many of these old ciphers are much to o weak to be used in applications to day, b ecause of the tremendous progress in computer technology. Until 1977 all ciphers were so-called one-key ciphers or conventional ciphers. In these ciphers the keys for encryption and decryption are identical or easily derived from each other. In 1977 Die and Hellman intro duced two-key ciphers or public-key ciphers, where the knowledge of one key gives no knowledge ab out the other key. In this pap er we consider only conventional ciphers. We divide these ciphers into three groups: Substitution Ciphers, Itansp osition Ciphers and Pro duct Ciphers. For each group we consider a sp eci c cipher and give a pictorial 1 illustration of an encryption. Finally we give an illustration that the strong conventional cipher, DES, in its basic mo de is insucient to ensure protection when used on a large plaintext. Notation: Let A and A b e the alphab ets for plaintexts and ciphertexts, resp ectively. M C Let M = m ;m ;:::;m be a n-character plaintext, s.t. for every i; m 2 0 1 n1 i A and let C = c ;c ;:::;c be a ciphertext, s.t. for everv i; c 2A M 0 1 n1 C 2 Substitution systems As indicated in the name every plaintext character is substituted by some ciphertext character. There are four kinds of substitution ciphers. Simple substitution Polyalphab etic substitution Homophonic substitution Polygram substitution We restrict ourselves to consider the rst two kinds. 2.1 Simple substitution In a cipher with a simple substitution each plaintext character is transformed into a ciphertext character via the same function f. More formally, 8i : 0 i<n f : A !A M C c =fm i i As an example the following 2 2.1.1 Caesar substitution It is b elieved that Julius Caessr encrypted messages by shifting every letter in the plaintext 3 p ositions to the right in the alphab et. This cipher is based on shifted alphabets, i.e. A = A , and is in M C general de ned as follows f m =m +k mo d jAMj i i For the Caesar cipher the secret key k is the number 3. In general, the cipher is easily broken in at most j AM j trials. Shift the ciphertexts one p osition until the Plaintext arises. 2.2 Polyalphab etic substitution In a p olyalphab etic substitution the plaintext characters are transformed into ciphertext characters using a j -character key K = k ;:::;k , which de nes j distinct functions F :::;F . More 0 j1 k k 0 j1 formally 8i :0<in f : A !A 8l :0l<j kl M C c =f m : i k i imo dj As an example the following 2.2.1 The Vigenere cipher The Vigenere cipher was rst published in 1586. In 1917 in an article in Scienti c American the cipher was claimed to b e \imp os- sible of translation" [Denning]. Let us assume again that A = A Then the Vigenere cipher is M C de ned as follows c = f m =m +k mo d jA j i k i i imo dj M imo dj Vigenere ciphers can b e broken when enough ciphertext is available to the cryptanalyst index of coincidence, Kasiski's metho d. 3 3 Transp osition systems Transp osition systems are essentially p ermutations of the plaintext characters. Therefore A = A . A transp osition cipher is de ned M C as follows 8i : O i<n f : A !A M M : f0;:::;n 1g! f0;:::;n 1g a p ermutation c : f m = m i i i Many transp osition ciphers p ermute characters with a xed p erio d j . In that case f : A !A M M : f0;:::;j 1g!f0;:::;j 1g a p ermutation c : f m = m i i idivj+ imo dj A convenient way to express the p ermutation i in easily memo- rable form is byakey word. The alphab etic order of the key characters then de nes the p ermutation. For example the key K=LARS would represent the p ermutation i = f1; 0; 2; 3g. Consider the following transp osition cipher 3.1 Row transp osition cipher Let the key be K = k ; ;k . The plaintext is divided into blo cks 1 d of d characters, and each blo ck is p ermuted according to the alphab etic order of the characters in the key. Let us consider an example: Example : Let d =4, the key K = IV AN and the plaintext M = NOTASTRONGCIPHER 4 I V A N 1 3 0 2 O A N T T O S R G I N C H R P E The ciphertext is C = OANTTOSRGINCHRPE We can do slightly b etter with the same key by combining row transp osition with columnar transp osition. Consider 3.2 Row and columnar transp osition ciphers Let the key be K = k ;:::;k . The plaintext is divided into blo cks 1 d of d d characters. Each blo ck is written into a d d matrix by rows according to the alphab etic order of the characters in the key. Then the ciphertexts is read from the matrix by columns according again to the alphab etic order of the characters in the key, thus the 2 p erio d of the cipher b ecomes d . Let us consider an example: Example : Let d =4, the key K=IVAN and the plaintext M = NOTASTRONGCIPHER M = NOTASTRONGCIPHER I V A N 1 3 0 2 I 1 O A N T V 3 T O S R G I N C A 0 N 2 H R P E 5 The ciphertext read out from the matrix is C = GINCOANTHRPETOSR l ansp osition ciphers can be broken, however, using tables of fre- quency distribution for digrams and trigrams. 4 Pro duct systems An obvious attempt to make stronger ciphers than the ones we've seen so far, is to combine substitution and transp osition ciphers. These ciphers are called pro duct cipher. Many pro duct ciphers has b een develop ed, including Rotor machines. We restrict ourselves to a short description of the most famous pro duct cipher ever develop ed. 4.1 Data Encryption Standard In 1977 15.01.77 the National Bureau of Standards in the U.S. published the Data Encryption Standard DES. The DES consists of 16 rounds of p ermutations and substitutlons. The DES encrypts 64 bit plaintexts into 64 bit ciphertexts using a 56 bit key. It Is beyond the scop e of this pap er to go further into details ab out the construction of the DES. We refer to the references in this pap er. The basic mo de of DES, the Electronic Co de Bo ok ECB Mo de, is de ned as follows c = DE S m i k i In other words c is the 64 bit encrypted value of the 64 bit plain- i text m i using the 56 bit key k . The next section illustrates, that this mo de is insucient to ensure protection when used on a large plaintext. For this purp ose the Cipher Blo ck Chaining CBC Mo de is recommended 6 c = DE S m c i k i i1 where c is a xed constant and is bitwise addition mo dulo 2. 1 The b est known attack on DES so far was published in 1991 by E. Biham and A. Shamir [Biham]. It nds the secret key, but 47 requires 2 chosen plaintexts. This is a very weak and unrealistic attack. In favour of the DES it should be noted, that the attacks on the ciphers mentioned in the preceding two Sections are known ciphertext attacks. These ciphers are trivially broken in a chosen plaintext attack. 5 The illustrations We consider the encryptions of a data- le 'FRAZETTA'. The le consists of 22080 characters of each 8 bits and is pictured in Figure 1. As can be seen the le contains redundancy. The Figures 2- 7 show the ciphertexts obtained by using the ciphers discussed earlier. The one-time pad is the only cipher, which is p erfect in the sense of Shannons theory. The cipher is de ned as follows c = m k i i i The obvious disadvantage is the long key. For 'FRAZETTA' we need a 22080 byte key. Figure 8 is a ciphertext obtained using the one-time pad and is included for comparison. 5.1 Comments on the gures Ad. Fig. 2-5: These four gures illustrates that the four ciphers do not provide sucient protection. The redundancy in the plaintext is more or less transferred to the ciphertexts. Ad. Fig. 6: In order to illustrate the inadequacy of the ECB Mo de only blo cks of 8 bits m are encrypted.

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